This is a little system I created inspired by the 13'th age background system. The point of the system is to allow characters to truly bind their backgrounds to their current character. It also allows some skill combinations that are hard to do with fixed skills or are really suboptimal. Take a former lion tamer, that character probably wants animal handling and performance, skills that are rarely used in my games and make the character weaker just because they want to follow the flavour of their character. I also wanted characters to have a little more skill "oomph" at low levels and reign in the +12-17 of high-level expertise. Hope you find it interesting.

At level 1 each character has three skillsets built around that character's backgrounds, both training and experience. Your starting bonuses for your skillsets are two with +3 and one with +2.
Each skillset is created "freeform", no set skills are used.
Skillsets are always subject to DM approval and its always the DM's choice of a skillset is applicable to a check or not.
Skillsets should not be so generic to apply to everything or so narrow to never be used, a character should have a useful skillset for around 50-60% of all ability checks
A good guideline is to have one fundament (court mage, sailor, priest, refuge, traveller) and one descriptor (in Waterdeep, of the astral sea, repentant, from the coming apocalypse, of the outback).

Whenever you would gain a skill beyond the standard 4 (2 from class, 2 from background) you may create an additional +2 skillset or add +1 to an existing skillset.
Whenever you reach level 3, 5, 7, 9, 11, 13, 15, 17 or 19 you may create an additional +2 skillset or add +1 to an existing skillset as a reflection of training and new experiences.
Rouges may at level 7, 13 and 19 create an extra +2 skillset or add an extra +1 to an existing skillset.
Bards may at level 7 and 13 create an extra +2 skillset or add an extra +1 to an existing skillset.
The maximum bonus in a skillset is +4. This increases to +5 at level 7 and to +6 at level 13.

Expertise: A expertise is more similar to an ordinary skill than a skillset, designating a specific thing your character is extraordinary capable in.
Whenever you roll a check you can apply your expertise to you make that roll with advantage.

Some examples:
Diplomat of Waterdeep might be useful for social checks, knowledge about Waterdeep, help to spot lies, gain access to nobility and more.
Disciple of the Archwizard Archial might help with arcane knowledge, magic items, the undead (that's a secret), history, socialising with wizards, dissecting animals and more.
A lvl 1 Half-elf Rogue has two +3 skillsets and one +2 with an additional 4 increases or additional skillsets to assign. (Max +4)

tldr: Start with two +3 skillsets and one +2. Add a +1 to a skillset every odd level or make another +2 skillset. Max bonus +4 at lvl 1-6, +5 at lvl 7-12 and +6 at lvl 13-20. Expertise grants advantage instead of dubbing.